On one boundary-value problem related to internal flotation

We study the problem of small motions of a system of immiscible ideal fluids with a free surface consisting of two domains: a domain of elastic ice and a domain of crushed ice. Elastic ice is modeled by an elastic plate. By crushed ice we mean weighty particles of some substance floating on the free surface. We also assume that the interface between the fluid layers is a weighty surface. Using the method of orthogonal projection of boundary conditions and the introduction of auxiliary problems, we reduce the original initial-boundary value problem to an equivalent Cauchy problem for a second-order differential equation in a Hilbert space. We obtain the conditions under which there is a strong-in-time solution of the initial-boundary value problem describing the evolution of this hydraulic system. We prove statements about the structure of the spectrum of the problem and about the basis property of the system of eigenfunctions.